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PreprintCC BY 4.0arXiv

Computation of Turing bifurcation normal form for n-component reaction-diffusion systems

  • Edgardo Villar-Sepúlveda
  • Alan R. Champneys
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Abstract

General expressions are derived for the amplitude equation valid at a Turing bifurcation of a system of reaction-diffusion equations in one spatial dimension, with an arbitrary number of components. The normal form is computed up to fifth order, which enables the detection and analysis of codimension-two points where the criticality of the bifurcation changes. The expressions are implemented within a Python package, in which the user needs to specify only expressions for the reaction kinetics and the values of diffusion constants. The code is augmented with a Mathematica routine to compute curves of Turing bifurcations in a parameter plane and automatically detect codimension-two points. The software is illustrated with examples that show the versatility of the method including a case with cross-diffusion, a higher-order scalar equation, and a four-component system.

Versions

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Cite

BibTeXRIS
BibTeX
@misc{villarsepulveda2026computation,
  title = {{Computation of Turing bifurcation normal form for n-component reaction-diffusion systems}},
  author = {Edgardo Villar-Sepúlveda and Alan R. Champneys},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.28039},
  eprint = {2609.28039},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.28039},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Computation of Turing bifurcation normal form for n-component reaction-diffusion systems
AU  - Edgardo Villar-Sepúlveda
AU  - Alan R. Champneys
PY  - 2026
DA  - 2026/09/23
PB  - arXiv
DO  - 10.48550/arxiv.2609.28039
UR  - https://arxiv.org/abs/2609.28039
AB  - General expressions are derived for the amplitude equation valid at a Turing bifurcation of a system of reaction-diffusion equations in one spatial dimension, with an arbitrary number of components. The normal form is computed up to fifth order, which enables the detection and analysis of codimension-two points where the criticality of the bifurcation changes. The expressions are implemented within a Python package, in which the user needs to specify only expressions for the reaction kinetics and the values of diffusion constants. The code is augmented with a Mathematica routine to compute curves of Turing bifurcations in a parameter plane and automatically detect codimension-two points. The software is illustrated with examples that show the versatility of the method including a case with cross-diffusion, a higher-order scalar equation, and a four-component system.
ER  -